On integrability of 2-dimensional $\sigma$-models of Poisson-Lie type

We describe a simple procedure for constructing a Lax pair for suitable 2-dimensional $\sigma$-models appearing in Poisson-Lie T-duality


Introduction
There is a class of 2-dimensional σ-models, introduced in the context of Poisson-Lie T-duality [5], whose solutions are naturally described in terms of certain flat connections. The target space of such a σ-model is D/H, where D is a Lie group and H ⊂ D a subgroup. The σ-model is defined by the following data: an invariant symmetric non-degenerate pairing , on the Lie algebra d such that the Lie subalgebra h ⊂ d is Lagrangian, i.e. h ⊥ = h, and a subspace V + ⊂ d such that dim V + = (dim d)/2 and such that , | V+ is positive definite. The construction and properties of these σ-models are recalled in Section 2 (including the Poisson-Lie Tduality, which says that the σ-model, seen as a Hamiltonian system, is essentially independent of H). Let us call them σ-models of Poisson-Lie type.
The solutions Σ → D/H of equations of motion of such a σ-model can be encoded in terms of d-valued 1-forms A ∈ Ω 1 (Σ, d) satisfying where V − := (V + ) ⊥ ⊂ d. Namely, the flatness (1a) of A implies that there is a map ℓ :Σ → D (whereΣ is the universal cover of Σ) such that A = −dℓ ℓ −1 . If the holonomy of A is in H then ℓ gives us a well-defined map Σ → D/H. The maps Σ → D/H obtained in this way are exactly the solutions of equations of motion. As first observed by Klimčík [3], and later by Sfetsos [12], and Delduc, Magro, and Vicedo [2], some σ-models of Poisson-Lie type are integrable. Their integrability is proven by finding a Lax pair, i.e. a 1-parameter family of flat connections (with parameter λ) A λ ∈ Ω 1 (Σ, g) dA λ + [A λ , A λ ]/2 = 0 where g is a suitable semisimple Lie algebra. Such a family is constructed for every element of the phase space, i.e. for every A ∈ Ω 1 (Σ, d) satisfying (1).
The aim of this note is to make the construction of A λ transparent. We simply observe that if A ∈ Ω 1 (Σ, d) satisfies (1) and if p : d → g is a linear map such that , p(A)]/2 = 0. A suitable family p λ : d → g will then give us a family of flat connections As an example, we provide a very simple construction of such families p λ in the case when d = g ⊗ W , where W is a 2-dimensional commutative algebra. These families recover the deformations of the principal chiral model from [2,3,12]. Our purpose is thus modest -it is simply to clarify previously constructed integrable σ-models. There is possibly a less naive construction of families p λ that might produce new integrable models, but we leave this question open.

σ-models of Poisson-Lie type and Poisson-Lie T-duality
In this section we review the properties of the "2-dimensional σ-models of Poisson-Lie type" introduced in [5] (together with their Hamiltonian picture from [6] and using the target spaces of the form D/H, as introduced in [7]).
Let d be a Lie algebra with an invariant non-degenerate symmetric bilinear form , of symmetric signature and let V + ⊂ d be a linear subspace with dim V + = (dim d)/2, such that , | V+ is positive-definite.
Let M = D/H where D is a connected Lie group integrating d and H ⊂ D is a closed connected subgroup such that its Lie algebra h ⊂ d is Lagrangian in d.
This data defines a Riemannian metric g and a closed 3-form η on M . They are given by and A ∈ Ω 1 (D, h) is the connection on the principal H-bundle p : D → D/H whose horizontal spaces are the right-translates of V + . 1 The metric g and the 3-form η then define a σ-model with the standard action functional where Σ is (say) the cylinder with the usual metric dσ 2 − dτ 2 and f : Σ → M is a map extended to the solid cylinder Y with boundary Σ.
For our purposes, the main properties of these σ-models are the following: • The solutions of the equations of motion are in (almost) 1-1 correspondence with 1-forms A ∈ Ω 1 (Σ, d) satisfying (1). More precisely, a map f : Σ → M is a solution iff it admits a lift ℓ :Σ → D such that A := −dℓ ℓ −1 satisfies (1). Notice that A is uniquely specified by f (the lift ℓ is not unique -it can be multiplied by an element of H on the right).
The d-valued functions j(σ) on the phase space of the sigma model satisfy the current algebra Poisson bracket Finally, let us observe that the phase space of the σ-model depends on the choice of H ⊂ D only mildly; when we impose the constraint that A has unit holonomy, the reduced Hamiltonian system is independent of H. This statement is the Poisson-Lie T-duality (in the case of no spectators). (In more detail, the phase space of the σ-model is the space of maps ℓ : R → D which are quasi-periodic in the sense that for some h ∈ H we have ℓ(σ + 2π) = ℓ(σ)h, modulo the action of H by right multiplication. The reduced phase space is (LD)/D (i.e. periodic maps modulo the action of D); it is the subspace of Ω 1 (S 1 , d) given by the unit holonomy constraint.)

Constructing new flat connections
As we have seen, the solutions of our σ-model give rise to flat connections A ∈ Ω 1 (Σ, d) satisfying (1). We can obtain new flat connections out of A using the following simple observation, which is also the main idea of this paper. Proposition 1. Let g be a Lie algebra and let p : d → g be a linear map such that Proof. Let us use the following notation: for α ∈ Ω 1 (Σ) let α + ∈ Ω 1,0 (Σ) and α − ∈ Ω 0,1 (Σ) denote the components of α, i.e. α = α + + α − . In particular, A + ∈ Ω 1,0 (Σ, V + ) and A − ∈ Ω 0,1 (Σ, V − ). We then have Given a 1-parameter family of maps p λ : d → g satisfying (4) we would thus get a 1-parameter family of flat connections A λ = p λ (A) on Σ, which may then be used to show integrability of the model. Let us observe that the Poisson brackets of the "Lax operators" L(σ, λ) := p λ (j(σ)) are automatically of the form considered in [10,11] (i.e. containing a δ(σ − σ ′ ) and a δ ′ (σ − σ ′ ) term), and so one can in principle extract an infinite family of Poisson-commuting integrals of motion out of the holonomy of A λ . Remark 1. The procedure of finding integrable deformations of integrable σmodels, due to Delduc, Magro, and Vicedo [1], can be rephrased in our formalism as follows. Suppose that for some particular pair V + ⊂ d we find a family p λ : d → g showing integrability of the model. Let us deform the Lie bracket on d, and possibly the pairing , , in such a way that the restriction of the Lie bracket to V + × V − → d is undeformed. Then the same family p λ will satisfy (4) also for the deformed structure on d and show integrability of the deformed model. These deformations of d do not change the system (1) (and if , is not deformed then they don't change the Hamiltonian (3) either), but they do change the Poisson structure (2) on the phase space.

Remark 2.
There is a version of σ-models of Poisson-Lie type, introduced in [8], with the target space if F \D/H, where f ⊂ d is an isotropic Lie algebra (and one needs to suppose that F acts freely on D/H). In this case V + ⊂ d is required to be such that , | V+ is semi-definite positive with kernel f (in particular, f ⊂ V + ), and such that [f, V + ] ⊂ V + (we still have dim V + = (dim d)/2). The phase space is the Marsden-Weinstein reduction of Ω 1 (S 1 , d) by LF , i.e. Ω 1 (S 1 , f ⊥ )/LF . The solutions of equations of motion are still given by the solutions of (1), though this time A is defined only up to F -gauge transformations. In this case we can still use Proposition 1 without any changes. This setup should cover, in particular, the discussion of symmetric spaces in [1].

Getting a Lax pair in a simple case
In this section we give a simple example of pairs V + ⊂ d with natural 1-parameter families p λ satisfying (4).
Let g be a Lie algebra with an invariant inner product , g and let W be a 2-dimensional commutative associative algebra with unit. (W is isomorphic to one of C, R ⊕ R, R[ǫ]/(ǫ 2 ).) Let We choose the following additional data in W to produce a pairing , on d and a subspace V + ⊂ d: To get the pairing, let θ : W → R be a linear form such that the pairing on W given by w 1 , w 2 W := θ(w 1 w 2 ) is non-degenerate (i.e. such that it makes W to a Frobenius algebra) and indefinite. The pairing on d is then defined via We can now describe the construction of a family p λ : d → g satisfying (4). Let us choose non-zero elements e + ∈ V 0 + and e − ∈ V 0 − (this choice is inessential). Proposition 2. If a linear form q : W → R satisfies (5) q(e + )q(e − ) = q(e + e − ) then the map p = id g ⊗ q : d → g satisfies (4).
The solutions q ∈ W * of (5) form a curve in W * , which is either a hyperbola or a union of two lines. If We thus have a hyperbola if ab = 0 and a union of two straight lines if ab = 0. One can easily check that ab = 0 iff one of V 0 ± is of the form Re where e ∈ W satisfies e 2 = e. This means that one of V ± = g⊗V 0 ± ⊂ d is a Lie subalgebra isomorphic to g and thus, according to [9], for any Lagrangian h ⊂ d, the corresponding σ-model is simply the WZW model given by G.
Let us now choose a rational parametrization λ → q λ of the hyperbola (5). The standard parametrization in this context seems to be the one sending λ = ±1 to the two points at the infinity of the hyperbola, and λ = ∞ to 0 (though any other parametrization would do). This gives (If the curve a union of two lines then this pametrizes only one of the lines, or possibly just a single point.) 0 (Σ, g) and A − ∈ Ω 0,1 (Σ, g) and if A is flat, then the g-connections The g-valued Lax operator obtained in this way is thus where we decomposed j(σ) as j(σ) = j + (σ) ⊗ e + + j − (σ) ⊗ e − . For completeness, the Hamiltonian (3) is

Examples of the example
In this section g is a compact Lie algebra and G the corresponding compact 1-connected Lie group.
Let start with the case of W = R ⊕ R, i.e. d = g ⊕ g. The only admissible θ ∈ W * , up to rescaling (which can be absorbed to , g ) and exchange of the two components of W , is θ(x, y) = x − y. (Here the main limiting factor is existence of a lagrangian Lie subalgebra h ⊂ d: if θ(x, y) = cx + dy with cd = 0 (the non-degeneracy condition), it forces c = −d.) The pairing , on d is ( We have e + = (1, t) e − = (t, 1) for some −1 < t < 1. The Lax operator (6), written in terms of j = (j 1 , j 2 ), is The Poisson brackets (2) of j 1,2 are The degenerate case t = 0 (when a = b = 0) corresponds to the WZW-model on g.
The natural choice for a Lagrangian Lie subalgebra h ⊂ d is the diagonal g ⊂ d. The target space of the σ-model is D/G ∼ = G. It is the so-called "λ-deformed σ-model" introduced by Sfetsos in [12] (Sfetsos's λ is our t).
Let us now consider the case W = C, which is the richest one. In this case any non-zero θ ∈ W * is suitable. Let θ(z) = Im(e 2iα z) for some α ∈ R. We thus have d = g ⊗ C = g C (seen as a real Lie algebra) with the pairing X, Y = Im(e 2iα X, Y g C ) where , g C is the C-bilinear extension of , g .
The Lax operator (6) is (where c.c. stands for "complex conjugate"). Here J = j re + ij im andJ = j re − ij im where j re and j im are given by j = j re ⊗ 1 + j im ⊗ i, their Poisson brackets (2) (written in an orthonormal basis of g) are The Hamiltonian (3) is 4 e 2iα J(σ), J(σ) g + e −2iα J (σ),J (σ) g dσ A suitable Lagrangian Lie subalgebra h ⊂ d can be found as follows. Let n ⊂ g C = d be the complex nilpotent Lie subalgebra spanned by the positive root spaces and let t ⊂ g be the Cartan Lie subalgebra. Let 0 = z ∈ C be such that θ(z 2 ) = 0; up to a real multiple we have z = e −iα or z = ie −iα . Then h = zt + n ⊂ g C = d is a real Lie subalgebra of d which is clearly Lagrangian. If z / ∈ R then h is transverse to g ⊂ d and we have an identification D/H ∼ = G for the target space of the σ-model.
The case of α = 0 corresponds to Klimčík's Yang-Baxter σ-model [3]. The general case is the Yang-Baxter σ-model with WZW term introduced in [2] and reinterpreted as a σ-model of Poisson-Lie type in [4].